What is the integral of ln(3x+8) ?

The integral of ln(3x+8) is xln(3x+8)+8/3 ln(3x+8)-x-8/3+C

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integral of ln(3x+8)

\int ln (3 x + 8) d x

x ln (3 x + 8) + \frac{8}{3} ln (3 x + 8) - x - \frac{8}{3} + C

Solution steps

\int ln (3 x + 8) d x

Apply u-substitution

: \int ln (u) \frac{1}{3} d u

= \int ln (u) \frac{1}{3} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{3} \cdot \int ln (u) d u

Apply Integration By Parts

: u ln (u) - \int 1 d u

= \frac{1}{3} (u ln (u) - \int 1 d u)

\int 1 d u = u

= \frac{1}{3} (u ln (u) - u)

Substitute back

u = 3 x + 8

= \frac{1}{3} ((3 x + 8) ln (3 x + 8) - (3 x + 8))

Simplify

\frac{1}{3} ((3 x + 8) ln (3 x + 8) - (3 x + 8)) : x ln (3 x + 8) + \frac{8}{3} ln (3 x + 8) - x - \frac{8}{3}

= x ln (3 x + 8) + \frac{8}{3} ln (3 x + 8) - x - \frac{8}{3}

Add a constant to the solution

= x ln (3 x + 8) + \frac{8}{3} ln (3 x + 8) - x - \frac{8}{3} + C

\frac{d}{d x} (\frac{1}{11 x ^{\frac{10}{11}}})

\int (x^{2} + 3) e^{x} d x

\frac{d}{d x} (arccos (2 x 1 - x^{2}))

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\int \frac{cos ( x )}{2} d x

What is the integral of ln(3x+8) ?
The integral of ln(3x+8) is xln(3x+8)+8/3 ln(3x+8)-x-8/3+C